Frequency tuning in RF generators is often used to reduce reflected power. A typical set-up is shown in FIG. 1. Typically, but not always, some type of matching network is used to match the load to the generator. By correct design of the matching network (either internal to the generator or external as shown in FIG. 1), it is possible to transform the impedance of the load to a value close to the desired load impedance of the generator (either at the RF output connector, typically 50Ω, or at the active devices internal to the generator, typically some low complex impedance such as 8+j3Ω) at some frequency in the range of frequencies that the generator can produce. The measure of how close the load impedance is to the desired impedance can take many forms, but typically it is expressed as a reflection coefficient
  ρ  =            Z      -              Z        0                    Z      +              Z        0        *            where ρ is the reflection coefficient of the impedance Z with respect to the desired impedance Z0 and x* means the complex conjugate of x. The magnitude of the reflection coefficient, |ρ|, is a very convenient way of expressing how close the impedance Z is to the desired impedance Z0. Both Z and Z0 are in general complex numbers.
Frequency tuning algorithms and methods try to find the optimal frequency of operation. Optimality is often defined as the frequency where the magnitude of the reflection coefficient with respect to the desired impedance is the smallest. Other measures may be minimum reflected power, maximum delivered power, stable operation etc. On a time-invariant linear load, many algorithms will work well, but on time-varying and/or nonlinear loads special techniques are required to ensure reliable operation of the tuning algorithm.
Assuming that the optimum frequency of operation is the frequency at which the load reflection coefficient magnitude is at its minimum, it is noted that the relationship between the controlled variable (frequency) and the error is frequently not monotonic and furthermore the optimum point of operation is generally at a point where the gain ([change in error]/[change in frequency]) is zero. To add to the challenges it is also possible that local minima may exist in which any control algorithm can get trapped. FIG. 2A shows a plot of load reflection coefficient on a load reflection coefficient chart (Smith chart) at the top, and FIG. 2B shows the magnitude of the load reflection coefficient used as the error as a function of frequency. This plot demonstrates the problems described above with a local minimum at f0 separated from the global optimum at fb by a region of high load reflection coefficient around fa and (as is invariably the case) zero slope of the error function at the global optimum frequency fb.
Two common problems on plasma loads are the nonlinear nature of the load (the load impedance is a function of power level) and that the load impedance changes over time (e.g., because of changing chemistry, pressure, temperature etc. over time). Another problem that is unique to plasma (or plasma-like) loads is that the plasma can extinguish if the delivered power to the plasma falls below some value for a long enough time. The frequency-tuning algorithm can therefore not dwell at a frequency where enough power cannot be delivered for very long or the plasma may extinguish.